While these approaches are very interesting and gave promising results, so far they have only been used to produce critical points for a fixed scalar potential, which is resulting from a single specific gauging within the large infinite family of possible deformations.
This leaves open the possibility that other vacua with the same residual symmetries appear in different gaugings.
The approach we are going to use in this work uses instead the power of the embedding tensor formalism in a way that allows for the search of critical points independently from the choice of gauging.
This approach was pioneered in a very different context in [1]} and used in the context of maximal 4-dimensional supergravity in [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]},
as well as in half-maximal supergravity in four and three dimensions [9]}, [10]}.
In addition to the power of investigating in a single sweep all deformations of maximal supergravity, this approach has so far produced analytic results for the critical points and their full spectrum, also providing information on the gauging, the residual gauge symmetry and supersymmetry of the vacua.
Moreover, for Minkowski vacua this led to understanding the moduli space of these theories [7]} as well as their uplift to string theory [12]}.
Finally, since the vacua are obtained without specifying first the gauging, this means that we can exhaustively classify vacua with a given residual symmetry for all possible consistent gaugings.
